Testing conditional multivariate rank correlations: the effect of institutional quality on factors influencing competitiveness

Joint distribution between two or more variables could be influenced by the outcome of a conditioning variable. In this paper, we propose a flexible Wald-type statistic to test for such influence. The test is based on a conditioned multivariate Kendall’s tau nonparametric estimator. The asymptotic properties of the test statistic are established under different null hypotheses to be tested for, such as conditional independence or testing for constant conditional dependence. Two simulation studies are presented: The first shows that the estimator proposed and the bandwidth selection procedure perform well. The second presents different bivariate and multivariate models to check the size and power of the test and runs comparisons with previous proposals when appropriate. The results support the contention that the test is accurate even in complex situations and that its computational cost is low. As an empirical application, we study the dependence between some pillars of European Regional Competitiveness when conditioned on the quality of regional institutions. We find interesting results, such as weaker links between innovation and higher education in regions with lower institutional quality. Supplementary Information The online version contains supplementary material available at 10.1007/s11749-022-00806-1.

For the expected value of theτ * z,hn estimator it holds that where I{Y 1 < Y 2 } = I{Y 11 < Y 12 , ..., Y p1 < Y p2 }. Applying the law of iterated expectations, the expected value above can be rewritten as follows: Applying expectations again, the above expression is . As a consequence, the asymptotic bias of the multivariate estimatorτ * z,hn is given by , the result still holds forτ z,hn .
Then, we derive the asymptotic distribution of theτ z,hn estimator through the convergence of the multivariate copula in a similar way to that already used for conditional copulas by Veraverbeke et al. (2011) and for right-censored length-biased data by Rabhi and Bouezmarni (2019), both in the bivariate case. In fact, it holds that I pĈz,hn (u)dĈ z,hn (u) = (nh n ) −1 n j=1 w j (z, h n )Ĉ z,hn (F z,hn (Y j )) provided that the left-hand side is the estimated mean ofĈ z,hn (u). Thus, substituting the copula estimatorĈ z,hn (u) in the latter expression leads to n i,j=1 w j (z, h n )w i (z, h n )I{Y i < Y j } + n j=1 w j (z, h n ) 2 . Hence, assuming that n j=1 w j (z, h n ) 2 = O ((nh n ) −1 ) holds,τ z,hn can be written aŝ Since Kendall's tau can be written as a functional of the copula, we define the map . In a similar way to Lemma 1 in Veraverbeke et al. (2011), ϕ is Hadamard differentiable at C z tangentially to the set of functions on [0, 1] p and its derivative is given by Hence, assuming h n = o(n −1/5 ), the delta method establishes that the asymptotic distribution ofτ z,hn is where C L z is the limiting distribution of (nh n ) −1/2 (Ĉ z,hn − C z ) so that the variance forτ z,hn is determined by σ 2 (ϕ ′ (C L z )). Note that h n = o(n −1/5 ) is required in the weak convergence of the empirical copula as well as to remove the conditional Kendall's tau estimator's bias.

Proof of Proposition 2
To derive the asymptotic distribution of the J n statistic we use the joint asymptotic normality of the conditional Kendall's tau given a set of covariate points.

Derumigny and Fermanian (2019) derive the asymptotic distribution of the conditional
Kendall's tau given a vector of covariates in a bivariate context. The generalization to our multivariate context for a unidimensional covariate can be obtained following same steps.
Proposition A.1. Let z = (z 1 , ..., z m ) ′ be m deterministic different points. Then, under assumptions A1 to A4, where Vτ z,hn is a matrix whose elements are defined by Proof of Proposition A.1.
Letτ z,hn the conditional Kendall's tau estimator defined by (3) Note thatτ z,hn is a multivariate version of the bivariateτ z estimator defined by Derumigny and Fermanian (2019), so Proposition A.1. can be proved using same arguments to those used to prove Proposition 9.
Given the nature of the conditional Kendall's tau as a functional of U-statistics, we define ϕ : (U n (g * ), U n (1)) −→ (U n (g * )/U n (1)) = (U n,ℓ (g * )/U n,ℓ (1)) ℓ=1,...,m , where its derivative is a m×2m order Jacobian matrix J ϕ (x, y) = [diag(y −1 1 , ...., y −1 m ), −diag(x 1 y −2 1 , ..., x m y −2 m )]. Hence, using the delta method over the map ϕ, it can be established that the joint limiting distribution ofτ z,hn is tuting the corresponding expressions and taking into account that all {z ℓ } m ℓ=1 are different, Vτ z,hn is a covariance matrix whose elements are given by To establish the limiting distribution of the test statistic, consider the asymptotic normality of the process (nh n ) 1/2 (τ z,hn −τ z ) given by Proposition A.1., Slutsky's theorem, and the properties of the normal distribution. Hence, (nh n ) 1/2 (Rτ z,hn −r) Since R is of full rank and Vτ z,hn is positive definite, RVτ z,hn R ′ is invertible. Therefore, we The asymptotic power of the test tends to the unity when h n → 0 and nh n → ∞ as n → ∞, as long as the alternative hypothesis holds. Note that if the linear restriction is getting closer and closer to the null hypothesis as the sample size increases, the power may not converge to unity. Therefore, to analyze the asymptotic power of the test, we consider local alternatives subject to Pitman sequences such that H a : R τ z = r + (nh n ) −1/2 ς. We remark that the convergence rate considered for the Pitman sequences is in line with the rates considered in the sequences of local alternatives for the tests based on U-statistics (see Zheng, 1996). The technique of using Pitman drifts to study the asymptotic power of t-ratio and Wald type statistics is also common in regression settings (see Hayashi, 2000).

Appendix B: Additional results
This section includes additional results to Section 3.